Diameter, Eccentricities and Distance Oracle Computations on H-Minor Free Graphs and Graphs of Bounded (Distance) Vapnik-Chervonenkis Dimension

Abstract

9 Under the Strong Exponential-Time Hypothesis, the diameter of general unweighted graphs 10 cannot be computed in truly subquadratic time (in the size n + m of the input), as shown 11 by Roditty and Williams. Nevertheless there are several graph classes for which this can be 12 done such as bounded-treewidth graphs, interval graphs and planar graphs, to name a few. We 13 propose to study unweighted graphs of constant distance VC-dimension as a broad generalization 14 of many such classes – where the distance VC-dimension of a graph G is defined as the VC-15 dimension of its ball hypergraph: whose hyperedges are the balls of all possible radii and centers 16 in G . In particular for any fixed H , the class of H -minor free graphs has distance VC-dimension 17 at most | V ( H ) | − 1. 18 • Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension 19 at most d , for any fixed k , either computes the diameter or concludes that it is larger than 20 k in time ˜ O ( k · mn 1 − ε d ), where ε d ∈ (0; 1) only depends on d 1 . We thus obtain a truly 21 subquadratic-time parameterized algorithm for computing the diameter on such graphs. 22 • Then as a byproduct of our approach, we get a truly subquadratic-time randomized algo-23 rithm for constant diameter computation on all the nowhere dense graph classes. The latter 24 classes include all proper minor-closed graph classes, bounded-degree graphs and graphs of 25 bounded expansion. Before our work, the only known such algorithm was resulting from 26 an application of Courcelle’s theorem, see Grohe et al. [47]. 27